Optimal. Leaf size=42 \[ \frac{3 x^{2/3}}{2}-3 \sqrt [3]{x}+6 \sqrt [6]{x}+3 \log \left (\sqrt [3]{x}+1\right )-6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]
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Rubi [A] time = 0.241455, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{3 x^{2/3}}{2}-3 \sqrt [3]{x}+6 \sqrt [6]{x}+3 \log \left (\sqrt [3]{x}+1\right )-6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]
Antiderivative was successfully verified.
[In] Int[(1 + Sqrt[x])/((1 + x^(1/3))*Sqrt[x]),x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ 2 \int ^{\sqrt{x}} \frac{\sqrt{x^{2}} + 1}{\sqrt [3]{x^{2}} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1+x**(1/2))/(1+x**(1/3))/x**(1/2),x)
[Out]
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Mathematica [A] time = 0.0183225, size = 42, normalized size = 1. \[ \frac{3 x^{2/3}}{2}-3 \sqrt [3]{x}+6 \sqrt [6]{x}+3 \log \left (\sqrt [3]{x}+1\right )-6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(1 + Sqrt[x])/((1 + x^(1/3))*Sqrt[x]),x]
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Maple [A] time = 0.006, size = 48, normalized size = 1.1 \[ \ln \left ( 1+x \right ) +{\frac{3}{2}{x}^{{\frac{2}{3}}}}+2\,\ln \left ( 1+\sqrt [3]{x} \right ) -\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{x}+1 \right ) -3\,\sqrt [3]{x}+6\,\sqrt [6]{x}-6\,\arctan \left ( \sqrt [6]{x} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1+x^(1/2))/(1+x^(1/3))/x^(1/2),x)
[Out]
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Maxima [A] time = 0.758112, size = 41, normalized size = 0.98 \[ \frac{3}{2} \, x^{\frac{2}{3}} - 3 \, x^{\frac{1}{3}} + 6 \, x^{\frac{1}{6}} - 6 \, \arctan \left (x^{\frac{1}{6}}\right ) + 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((sqrt(x) + 1)/(sqrt(x)*(x^(1/3) + 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.280949, size = 41, normalized size = 0.98 \[ \frac{3}{2} \, x^{\frac{2}{3}} - 3 \, x^{\frac{1}{3}} + 6 \, x^{\frac{1}{6}} - 6 \, \arctan \left (x^{\frac{1}{6}}\right ) + 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((sqrt(x) + 1)/(sqrt(x)*(x^(1/3) + 1)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 18.2063, size = 39, normalized size = 0.93 \[ 6 \sqrt [6]{x} + \frac{3 x^{\frac{2}{3}}}{2} - 3 \sqrt [3]{x} + 3 \log{\left (\sqrt [3]{x} + 1 \right )} - 6 \operatorname{atan}{\left (\sqrt [6]{x} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1+x**(1/2))/(1+x**(1/3))/x**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.261293, size = 41, normalized size = 0.98 \[ \frac{3}{2} \, x^{\frac{2}{3}} - 3 \, x^{\frac{1}{3}} + 6 \, x^{\frac{1}{6}} - 6 \, \arctan \left (x^{\frac{1}{6}}\right ) + 3 \,{\rm ln}\left (x^{\frac{1}{3}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((sqrt(x) + 1)/(sqrt(x)*(x^(1/3) + 1)),x, algorithm="giac")
[Out]